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Theorem xrninxp2 34474
Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)
Assertion
Ref Expression
xrninxp2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝐶,𝑥   𝑢,𝑅,𝑥   𝑢,𝑆,𝑥

Proof of Theorem xrninxp2
StepHypRef Expression
1 inxp2 34452 . 2 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)}
2 3ancoma 1084 . . . 4 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢𝐴𝑢(𝑅𝑆)𝑥))
3 df-3an 1074 . . . 4 ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢(𝑅𝑆)𝑥) ↔ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥))
4 3anass 1081 . . . 4 ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑢𝐴𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
52, 3, 43bitr3i 290 . . 3 (((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥)))
65opabbii 4869 . 2 {⟨𝑢, 𝑥⟩ ∣ ((𝑢𝐴𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅𝑆)𝑥)} = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
71, 6eqtri 2782 1 ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢𝐴𝑢(𝑅𝑆)𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1072   = wceq 1632  wcel 2139  cin 3714   class class class wbr 4804  {copab 4864   × cxp 5264  cxrn 34295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274
This theorem is referenced by:  inxpxrn  34476
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